Using trigonometry to solve problems in the real world involves not only identifying the relevant triangles, but also knowing which trigonometric tools to use to get the information we want. Being able to choose the correct tool is an important skill for solving any problem. Lets have another look at the tools we can now use:
The last three of these can be used for any triangle, however the first two in the list depend on the triangle having a right-angle.
Remember, these tools are only able to be used in triangles which are right-angled.
For the triangle shown, the following equations are always true:
$a^2+b^2=c^2$a2+b2=c2
and
$\sin\theta=\frac{a}{c},\cos\theta=\frac{b}{c},\tan\theta=\frac{a}{b}$sinθ=ac,cosθ=bc,tanθ=ab
The following table describes what which right-angled triangle rules you can use and how, given the information you already know. Notice in these cases that we always know the size of at least one angle, which is the right-angle.
These rules describe the relationship between different sides and angles of a triangle, whether the triangle has a right-angle or not.
For the triangle shown, the following equations are always true:
$\frac{\sin A}{a}$sinAa$=$=$\frac{\sin B}{b}=\frac{\sin C}{c}$sinBb=sinCc (the sine rule)
and
$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC (the cosine rule)
As we've noticed in previous chapters, each rule can be used in different situations because they rely on different information to solve for an angle size or side length. When you see a triangle and want to solve for the information you don't know, the first step is to figure out what you do know about the triangle. Then, choose the rule which is able to use that information.
The following table describes what rules you can use and how, given the information you know about a triangle.
Like the sine and cosine rules, the area rule can be used for any triangle.
For the triangle shown, the following equation is always true:
$Area$Area$=$=$\frac{1}{2}ab\cos C$12abcosC
Since this is the only rule so far that relates the side lengths and angles of a triangle to its area, this rule will be helpful whenever the area of a triangle is unknown.
$\triangle ABC$△ABC consists of angles $A$A, $B$B and $C$C which appear opposite sides $a$a, $b$b and $c$c respectively. Consider the case where the measures of $a$a, $b$b and $A$A are given.
Which of the following is given?
Two angles and the side between them
Two sides and the included angle
Three sides
Two sides and an angle
Two angles and one side
Which of the following should be used to begin solving for the remaining sides and angles in the triangle?
The sine rule
Pythagoras' theorem
The area rule
The cosine rule
Calculate the length of $y$y in metres.
Round your answer to one decimal place.
Find the value of angle $w$w in degrees.
Round your answer to two decimal places.